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I am currently reading Roger Penrose's The Road to Reality and in the book, the author poses various problems with three different levels of difficultly easy, hard and really hard, according to the author this is easy. The problem I am looking at is as follows:

Using the power series of $e^x$ show that $de^x = e^x \, dx$

I have no idea as to how to tackle this problem.

If someone could provide some key points to solving the problem that would be great. Please do not provide the full steps, just key ideas or things to note.

Thanks!

EDIT:

I believe I understand this now because when you take the derivative of a power series you can do it term by term. The power series for $e^x$ is:

$$e^x = \sum_{i=0}^\infty \frac{x^n}{n!}$$

But more expanded it looks like this:

$$e^x = \sum_{i=0}^{\infty} 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} +\ldots $$

If the derivative of each term then I get: $0 + 1 + x + \frac{x^2}{2} + \ldots$ So in essence, I'm coming back to the original series. Therefore, the derivatives are the same.

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That's about right. But no one really explained why the fact that it is a power series implies that you can differentiate term by term. Do you know why? –  Git Gud Mar 2 '13 at 19:27
    
@GitGud No I don't, can you just think of it as a polynomial that is begin added? Like one that looks like this: $a_nx^n + a_{n-1}x^n-1 \ldots$ but more compact? And that is why you can do it term for term. If it it was $\prod$ instead of $\sum$ we would have to use the product rule instead of the power rule right? If my initial assertion is correct. –  Jeel Shah Mar 2 '13 at 19:29
    
Do you know anything about function series? –  Git Gud Mar 2 '13 at 19:32
    
@GitGud Not really. I understand the basic idea behind $\sum$ and how to do some basic things with it but I don't know much beyond that but is what I said correct? Like the basic idea of it? –  Jeel Shah Mar 2 '13 at 19:33
    
Then I think you're not supposed to think about why it is legal to differentiate term by term. It's Physics anyway, so just do it. –  Git Gud Mar 2 '13 at 19:35

4 Answers 4

up vote 6 down vote accepted

The power series representation of $e^x$ is $$ e^x = \sum_{i=0}^{\infty} \frac{x^n}{n!} $$ When you take the derivative of a power series you can do it term by term: $$\begin{align} \frac{d}{dx}e^x &= \frac{d}{dx}\sum_{i=0}^{\infty} \frac{x^n}{n!} \\ &= \sum_{i=0}^{\infty} \frac{d}{dx}\frac{x^n}{n!} \\ &= \sum_{i=1}^{\infty}\frac{x^{n-1}}{(n-1)!} \end{align} $$ This is the same as $e^x$ (think about it).

To add a bit more detail. Remember that there is a rule that says that $$ \frac{d}{dx} x^n = nx^{n-1}. $$ You for example have that $$ \frac{d}{dx} x^2 = 2x \quad\quad\frac{d}{dx} x^3 = 3x^2. $$ So in the above we have used that $$\frac{d}{dx}\frac{x^n}{n!} =\frac{1}{n!} \frac{d}{dx} x^n = \frac{1}{n!}nx^{n-1} = \frac{n}{n\cdot(n-1)!}x^{n-1} = \frac{x^{n-1}}{(n-1)!}. $$

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I have edited the post. Is my understanding correct? The fact that when you find each derivative, you will get the answer as $n-1$ term of the original series? –  Jeel Shah Mar 2 '13 at 19:27
    
@gekkostate: What you say in your edited question is correct. So the derivative of each term gives the term just before it. So in all you end up with the same series. –  Thomas Mar 2 '13 at 19:35

Just write it out,

$$e^x=\sum_{n\ge 0}\frac{x^n}{n!}=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\ldots\;,$$

and differentiate it term by term. What’s the derivative of $\dfrac{x^n}{n!}$?

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I have not yet encountered how to find the derivative of $\frac{x^n}{n!}$, we haven't been taught this in school. What's the idea behind find that derivative? –  Jeel Shah Mar 2 '13 at 19:09
1  
@gekkostate: Do you know how to find the derivative of $x^n$? That’s pretty much a prerequisite for answering the question. –  Brian M. Scott Mar 2 '13 at 19:11
    
@gekkostate I think you know how to differentiate that. You're probably confusing something. Can you find the derivatives of $\displaystyle x, \frac{x^2}{2}$ and $\displaystyle \frac{x^3}{6}$? –  Git Gud Mar 2 '13 at 19:11
1  
@gekkostate: $n!$ is just a number. When you differentiate $\dfrac{x^3}{3!}$, you’re just differentiating $\dfrac{x^3}6$, so you get $\dfrac{x^2}2$. But it’s more useful here to think of that as differentiating $\dfrac{x^3}{3\cdot2\cdot1}$ and getting $\dfrac{3x^2}{3\cdot2\cdot1}=\dfrac{x^2}{2\cdot1}$. –  Brian M. Scott Mar 2 '13 at 19:15
2  
@gekkostate note that $n\in \Bbb N$ is already a fixed constant. Therefore $\displaystyle d\frac{x^n}{n!}=\frac{1}{n!}dx^n$. And to answer your question: yes, it is the same principle. –  Git Gud Mar 2 '13 at 19:17

We denote $\displaystyle f_n(x)=\frac{x^n}{n!}$, so we have $\displaystyle e^x=\sum_{n=0}^{\infty}f_n(x).$

Let $[a,b]$ an arbitrary interval.

To justify the term by term differntiation of the series on $[a,b]$ we verify this points:

  • the series $\displaystyle \sum_{n=0}^{\infty}f_n(x_0)$ at some point $x_0\in[a,b]$ and
  • the series of derivatives $\displaystyle \sum_{n=0}^{\infty}f'_n(x)$ converges uniformly on $[a,b]$ to, say, $g$,

in this case, the series $\displaystyle \sum_{n=0}^{\infty}f_n(x)$ converges at every $x\in [a,b]$ and $$\displaystyle \left(\sum_{n=0}^{\infty}f_n(x)\right)'=g(x).$$

In our question, the two points are straightforward, for instance, we verify the second point: we have $$|f'_n(x)|=\frac{|x|^{n-1}}{(n-1)!}\leq \frac{\max(|a|,|b|)^{n-1}}{(n-1)!}=c_n,$$ so the series $\displaystyle\sum_nf'_n(x)$ converges uniformly on $[a,b]$ by Weierstrass M-test since the series $\displaystyle\sum_n c_n$ is convergent.

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If you are having trouble to visualize this series with sum notation, just write down term by term and derive them. Doing this you will get the idea.

$\displaystyle e^x=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\ldots$

$\displaystyle \frac{d}{dx}e^x=\frac{d}{dx}\big(1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\ldots\big)=$

$\displaystyle =\frac{d}{dx}\big(1\big)+\frac{d}{dx}\big(\frac{x}{1!}\big)+\frac{d}{dx}\big(\frac{x^2}{2!}\big)+\frac{d}{dx}\big(\frac{x^3}{3!}\big)+\frac{d}{dx}\big(\frac{x^4}{4!}\big)+\ldots=$

$\displaystyle =0+\frac{1}{1!}\frac{d}{dx}(x)+\frac{1}{2!}\frac{d}{dx}(x^2)+\frac{1}{3!}\frac{d}{dx}(x^3)+\frac{1}{4!}\frac{d}{dx}(x^4)+\ldots=$

$\displaystyle =\frac{1}{1!}+\frac{1}{2!}2x+\frac{1}{3!}3x^2+\frac{1}{4!}4x^3+\ldots=$

$\displaystyle =\sum_{n=1}^\infty \frac{nx^{n-1}}{n!}=\sum_{n=1}^\infty \frac{x^{n-1}}{(n-1)!}$

I hope this clarify all the process done.

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We can differentiate each term separately because without the $\sum$, the function would be like any other polynomial of the form $a_nx^n + a_{n-1}x^{n-1} \ldots a_1 + a_0$ and that is why we can use the power rule on each term right? Is that the logical behind being able to differentiate each term separately? –  Jeel Shah Mar 2 '13 at 19:45
    
You can think that way because your teacher probably will make you solve only "good" series. But the reason is a little more deep. –  Integral Mar 2 '13 at 19:51
    
hmm, okay. Thanks for your answer! –  Jeel Shah Mar 2 '13 at 19:54
1  
If you want to understand all the details, you'll need to study real analysis. –  Integral Mar 2 '13 at 19:54

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